1. \[ y’ = -\operatorname{cosech} x \]
2. \[ y’ = 2 \cosh 2x e^{\operatorname{senh} 2x} \]
3. \[ y’ = x^{\tanh x} \left( \frac{\tanh x}{x} + \operatorname{sech}^2 x \ln x \right) \]
4. \[ y’ = \frac{1}{4} \operatorname{sech}^4 \frac{x}{2} \]
5. \[ y’ = e^{ax} [a \cosh bx + b \operatorname{senh} bx] \]
6. \( y’ = \cfrac{1}{2} \sqrt[4]{\cfrac{1+\tanh x}{1-\tanh x}}\) \( = \cfrac{1}{2\sqrt{\cosh x – \operatorname{senh} x}} \)
7. \[ y’ = \frac{-2 \operatorname{cosech}^{-1} x}{|x|\sqrt{1+x^2}} \]
8. \[ y’ = \frac{2x}{\sqrt{x^4+a^4}} \]
9. \[ y’ = -\operatorname{cosec} x \]
10. \[ y’ = \frac{2}{1-x^4} \]

En los problemas del 1 al 10, hallar la derivada \( y’ = D_x y \).

1. \[ y = \tanh^{-1}(\cosh x) \]
2. \[ y = e^{\operatorname{senh}(2x)} \]
3. \[ y = x^{\tanh x}, \quad x > 0 \]
4. \[ y = \frac{1}{2} \tanh \left(\frac{x}{2}\right) – \frac{1}{6} \tanh^3 \left(\frac{x}{2}\right) \]
5. \[ y = e^{ax} \cosh bx \]
6. \[ y = \sqrt[4]{\frac{1+\tanh x}{1-\tanh x}} \]
7. \[ y = (\operatorname{cosech}^{-1} x)^2 \]
8. \[ y = \operatorname{senh}^{-1} \frac{x^2}{a^2} \]
9. \[ y = \tanh^{-1}(\sec x) \]
10. \[ y = \tan^{-1} x + \tanh^{-1} x \]

11. Probar las siguientes identidades dadas en el teorema 3.5.1:

    1. \( \operatorname{senh}(-x) = -\operatorname{senh} x \)
    2. \( \cosh(-x) = \cosh x \)
    3. \( 1 – \coth^2 x = -\operatorname{cosech}^2 x \)
    4. \( \cosh(x + y) = \cosh x \cosh y\) \(+ \operatorname{senh} x \operatorname{senh} y \)
12. Probar las identidades:

    1. \( \operatorname{senh}(x – y) = \operatorname{senh} x \cosh y\) \(- \cosh x \operatorname{senh} y \)
    2. \( \cosh(x – y) = \cosh x \cosh y\) \(- \operatorname{senh} x \operatorname{senh} y \)
    3. \( \cosh x + \cosh y = 2 \cosh \frac{x+y}{2} \cosh \frac{x-y}{2} \)
    4. \( \operatorname{senh} x + \operatorname{senh} y = 2 \operatorname{senh} \frac{x+y}{2} \cosh \frac{x-y}{2} \)
    5. \( \cosh x – \cosh y = 2 \operatorname{senh} \frac{x+y}{2} \operatorname{senh} \frac{x-y}{2} \)
    6. \( \operatorname{senh} 3x = 3 \operatorname{senh} x + 4 \operatorname{senh}^3 x \)
13. Probar las igualdades 2, 3, 4, 5 y 6 del teorema 3.5.4.